Name.........................................................................................
I.D. number................................................................................
Math 2250-1
Exam 1
October 6, 2008
This exam is closed-book and closed-note. You may use a scientific calculator, but not one which is
capable of graphing or of solving differential or linear algebra equations. In order to receive full or
partial credit on any problem, you must show all of your work and justify your conclusions. There
are 100 points possible. The point values for each problem are indicated in the right-hand margin.
Good Luck!
Score:
Possible
1___________
35
2___________
10
3___________
20
4___________
10
5___________
25
Total____________
100
1) Consider a brine tank which holds 1000 gallons of continuously-mixed liquid. Let x(t) be the amount
of salt (in pounds) in the tank at time t. The in-flow and out-flow rates are both 200 gallons/hour, and
the concentration of salt flowing into the tank is 0.05 pounds per gallon.
1a) Use the information above and your modeling ability to derive the differential equation for x(t):
dx
= 10 0.2 x
dt
(5 points)
1b) What is the equilibrium solution to this differential equation? Is it stable or unstable? Explain using
a phase portrait.
(5 points)
1c) Find the solution to the initial value problem for this differential equation,
dx
= 10 0.2 x ,
dt
assuming there are originally 30 pounds of salt in the tank. This is a linear differential equation, so use
the algorithm you learned for this sort of DE.
(10 points)
1d) The differential equation we are considering in this problem is also separable. Use the algorithm
you learned for separable DEs to resolve the initial value problem in (1c).
(10 points)
1e) For the solution you find in (1cd), is tlim x t consistent with your discussion in (1b)? How could
your smart little sister have figured out the limiting amount of salt in the tank without understanding any
differential equations, i.e. just by using the in-flow concentration?
(5 points)
2a) Your father has a "sporty" car of mass 1000 kg and the engine and transmission of this car is able to
kg m
provide a maximum force of 10,000 Newtons ( 2 ) at all speeds. Assume the frictional forces on the
sec
car from air and road resistance are proportional to the car’s velocity, i.e. we are studying a linear drag
model. What is the drag coefficient, and what are its units, if the velocity of the car satisfies
dv
= 10 0.2 v ?
dt
(5 points)
2b) Notice that you’ve ended up with the same differential equation as in problem 1 on this test!
Suppose you’re driving your Dad’s car at a raceway, going 30 m/sec, and you begin to accelerate at the
maximum rate possible. How long will it until you are going 45 m/sec (about 101 miles per hour)? Use
your solution from 1d) or 1e)!
(5 points)
3) Consider the following matrix equation, of the form Ax=b:
2 1 2
x
6
1
1 1
y =
3
1
5 1
z
3
3a) Compute the determinant of the coefficient matrix A above. What this tells you about the matrix
equation and its possible solutions. In particular, can you rule out any of the three general possibilities:
one solution, no solutions, or infinitiely many solutions, based on your determinant computation?
Explain.
(8 points)
3b) Compute the reduced row echelon form of the 3 by 4 augmented matrix associated to the matrix
equation above, and use it to find all solutions to the system.
(12 points)
4) Use the adjoint formula for the inverse matrix or Cramer’s rule, to solve the following small but
messy system of equations:
5x
7x
11 y = 2
16 y = 3
(10 points)
5) Consider the differential equation
dP
= P2 2 P
dt
which could be a model for a certain population problem.
5a) Find the equilibrium solutions.
(5 points)
5b) Sketch the phase portrait for this autonomous DE and decide whether each equilibrium solution is
stable or unstable.
(5 points)
5c) Which of population models we studied would lead to differential equation of this type? Be as
precise as you can, so that you account for the signs of both terms on the right of the differential
equation.
(5 points)
5d) Solve the initial value problem
dP
= P2 2 P
dt
P 0 =3
Explain what happens to your solution as time increases, and whether this behavior is consistent with
the phase portrait you drew in (5b).
(10 points)